Towards multireference equivalents of the G2 and G3 methods

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1 Towards multireference equivalents of the G2 and G3 methods Theis I. So/lling, David M. Smith, Leo Radom, Mark A. Freitag, and Mark S. Gordon Citation: The Journal of Chemical Physics 115, 8758 (2001); doi: / View online: View Table of Contents: Published by the AIP Publishing Articles you may be interested in Intersystem-crossing and phosphorescence rates in fac-ir III (ppy)3: A theoretical study involving multi-reference configuration interaction wavefunctions J. Chem. Phys. 142, (2015); / A theoretical study of the excited states of Am O 2 n +, n = 1, 2, 3 J. Chem. Phys. 128, (2008); / A convenient decontraction procedure of internally contracted state-specific multireference algorithms J. Chem. Phys. 124, (2006); / The X Σ g + 1, B Δ g 1, and B Σ g + 1 states of C 2 : A comparison of renormalized coupled-cluster and multireference methods with full configuration interaction benchmarks J. Chem. Phys. 122, (2005); / Ab initio study of the electron-spin magnetic moments (g-factors) of C 2, CSi, Si 2, LiC 2, NaC 2, and LiSi 2 J. Chem. Phys. 112, (2000); /

2 JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER NOVEMBER 2001 Towards multireference equivalents of the G2 and G3 methods Theis I. So lling, David M. Smith, and Leo Radom a) Research School of Chemistry, Australian National University, Canberra, ACT 0200, Australia Mark A. Freitag and Mark S. Gordon b) Department of Chemistry, Iowa State University, Ames, Iowa Received 2 May 2001; accepted 24 August 2001 The effect of replacing the standard single-determinant reference wave functions in variants of G2 and G3 theory by multireference MR wave functions based on a full-valence complete active space has been investigated. Twelve methods of this type have been introduced and comparisons, based on a slightly reduced G2-1 test set, are made both internally and with the equivalent single-reference methods. We use CASPT2 as the standard MR-MP2 method and MRCl Q asthe higher correlation procedure in these calculations. We find that MR-G2 MP2,SVP, MR-G2 MP2, and MR-G3 MP2 perform comparably with their single-reference analogs, G2 MP2,SVP, G2 MP2, and G3 MP2, with mean absolute deviations MADs from the experimental data of 1.41, 1.54, and 1.23 kcal mol 1, compared with 1.60, 1.59, and 1.19 kcal mol 1, respectively. The additivity assumptions in the MR-Gn methods have been tested by carrying out MR-G2/MRCI Q and MR-G3/MRCI Q calculations, which correspond to large-basis-set MRCI Q ZPVE HLC calculations. These give MADs of 1.84 and 1.58 kcal mol 1, respectively, i.e., the agreement with experiment is somewhat worse than that obtained with the MR-G2 MP2 and MR-G3 MP2 methods. In a third series of calculations, we have examined pure MP2 and MR-MP2 analogs of the G2 and G3 procedures by carrying out large-basis-set MP2 and CASPT2 ZPVE HLC calculations. The resultant methods, which we denote G2/MP2, G3/MP2, MR-G2/MP2, and MR-G3/MP2, give MADs of 4.19, 3.36, 2.01, and 1.66 kcal mol 1, respectively. Finally, we have examined the effect of using MCQDPT2 in place of CASPT2 in five of our MR-Gn procedures, and find that there is a small but consistent deterioration in performance. Our calculations suggest that the MR-G3 MP2 and MR-G3/MP2 procedures may be useful in situations where a multireference approach is desirable American Institute of Physics. DOI: / I. INTRODUCTION The prediction of thermodynamic properties, such as atomization energies, ionization energies, electron affinities, and heats of formation, to chemical accuracy has long been a goal of quantum chemists, and there has been great progress in this direction in recent years. 1 Methods that have been developed in an attempt to achieve this goal, include the Gaussian series Gn, n 1,2or3 of model chemistries developed by Curtiss, Raghavachari, Pople, and co-workers, 2 4 the complete-basis-set CBS methods of Petersson and co-workers, 5 the BAC-MPX X 2 or4 methods due to Melius and co-workers, 6 the W1, W2 and related methods of Martin, 7 and the extrapolation procedures due to Dunning, Feller, Dixon, Peterson, and co-workers. 8 The G2 and G3 methods and their variants, 3,4 in particular, have become very popular among both theoreticians and expermentalists, because of their ability to predict accurate thermodynamics for a wide variety of chemical compounds. One potential drawback of the Gn approaches is that they are based on the presumption that the chemical species of interest can be well described by a single configuration, i.e., it can be well represented by a single Lewis structure. There are, however, many systems for which this assumption may not be appropriate. 9 Important examples include transition structures for many chemical reactions, regions of potential energy surfaces in which bonds are dissociating or forming near conical intersections, as well as the vast majority of electronic excited states. 9 In addition, first-row transition metal complexes and unsaturated compounds that contain transition metals are also often not well described by a single-determinant wave function. For such species with pronounced multireference character, the Gn methods may not provide accurate thermodynamic quantities. 9 The aim of the various Gn models is generally to estimate energies at a high correlation level, typically quadratic configuration interaction QCISD T, 10 with a large basis set. This is achieved by starting with a modest-basis-set QCISD T calculation and estimating the effect of moving to a larger basis set at the MP2 and /or MP4 levels, i.e., assuming the additivity of basis set and correlation effects. In addition, a zero-point vibrational energy correction is incorporated, as well as a higher level correction, which is intended to account for any remaining deficiencies in level of theory and basis set. In a multireference MR Gn approach, the analog of MP2 would be MR-MP2 while the analog of QCI would ideally be MR-QCI. Unfortunately, codes for carrying out MR-QCI or related coupled cluster MR-CC cala Electronic mail: radom@rsc.anu.edu.au b Electronic mail: mark@si.fi.ameslab.gov /2001/115(19)/8758/15/$ American Institute of Physics

3 J. Chem. Phys., Vol. 115, No. 19, 15 November 2001 Multireference G2 and G TABLE I. Higher-level-correction parameters in mhartrees for the MR-G2 and G2-type methods. TABLE II. Higher-level-correction parameters in mhartrees for the MR-G3 and G3-type methods. Method A Method A B C D MR-G2 MP2.SVP MR QD -G2 MP2.SVP MR-G2 MP MR QD -G2 MP MR-G2/MRCI Q MR-G2/MP MR QD -G2/MP G2/MP MR-G3 MP MR QD -G3 MP MR-G3/MRCI Q MR-G3/MP MR QD -G3/MP G3/MP culations are not widely available at the present time. We have selected multireference configuration interaction with single and double substitutions MR-CISD as the best current alternative. The present multireference analogs of both the G2 and G3 methods are described in Sec. II. It is important as a first step in developing MR-Gn procedures that may be usefully applied to problems requiring a multireference approach, to examine the performance of such procedures for systems that are reasonably described by single-reference treatments. That is the principal aim of the present study and the G2-1 test set is well suited for this purpose. Accordingly, since the new multireference procedures are essentially untested with respect to their ability to reliably predict accurate thermodynamic quantities in the manner of the Gn methods, Sec. III contains a detailed assessment of their performance on a slightly modified G2-1 test set. We also examine the performance of pure MP2 and MR-MP2 analogs of G2 and G3 theory. Finally, conclusions that emerge from our study are presented in Sec. IV. set of MR-Gn procedures, we retain the same geometries MP2 full /6-31G(d) and ZPVEs scaled HF/6-31G(d) as in the Gn methods and these are thus taken from the G2 data base. 11 This makes it easier to identify inherent MR-Gn differences. The current single-configuration levels of theory are replaced by multireference analogs as follows: SCF MCSCF, 2 MP2 MR-MP2, QCISD T MR-CISD. MCSCF refers to multiconfiguration MC self-consistentfield SCF calculations based on the CASSCF Ref. 12 or FORS Ref. 13 prescription. We include all valence electrons and valence orbitals in the active space. For example, the active spaces for methane, ammonia, and water are 8, 8, 8, 7, and 8, 6, respectively, where the first number is the number of active electrons and the second number refers to the number of active orbitals. By choosing a full-valence CASSCF approach, we obtain a procedure that is well-defined for any species, but the downside is that the cost rises very rapidly with molecular size. Our standard II. METHODS MR-MP2 multireference second-order perturbation theory A. Relationship between the Gn and MR-Gn methods method is the CASPT2 procedure developed by Roos and co-workers. 14,15 We also examine results obtained with the The simplest version of the G2 method, referred to as G2 MP2,SVP, 3 e,3 f multiconfiguration quasidegenerate second-order perturbation theory method, MCQDPT2, developed by Nakano. 16 We is based on the following additivity approximation to estimate the QCISD T energy for the extended G 3df,2p) basis set, note that analytic gradients for MCQDPT2 have been derived, also by Nakano, 17 and are currently being implemented into the electronic structure code GAMESS. 18 This E QCISD T /6-311 G 3df,2p may be important in more refined versions of MR-Gn in E QCISD T /6-31G d which the geometries are reoptimized at MR-MP2 rather E MP2/6-311 G 3df,2p E MP2/6-31G d. than simply using the MP2 geometries of Gn theory. The remaining step in the MR-Gn model requires a 1 multi-reference energy calculation at a level of theory that is The G2 MP2,SVP energy is derived by adding to this, comparable to QCISD T. The obvious choice would be firstly a zero-point vibrational energy ZPVE obtained from MR-QCISD T. While several groups have worked on multireference coupled cluster methods, 19 there are no efficient, scaled by HF/6-31G(d) vibrational frequencies, and secondly a higher level correction HLC. The HLC is an general MR-CCSD T codes available at the present time. empirical correction which is determined by minimizing the So, while in the long term it is desirable to use MRmean absolute deviation MAD between experiment and QCISD T or MR-CCSD T for this step of the method, in theory for the thermochemical quantities in a test set of molecules see below. CISD of Werner and Knowles 20 with the Davidson cluster the short term we will use the internally-contracted MR- The multireference versions of the Gn schemes are correction Q. We will refer to this method as MRCI Q based on the same premise as the single-reference version, throughout the present work. namely, that the effects of improvements in the basis set and In this manner, we have constructed the MR-G2-type treatment of electron correlation are additive. In our initial and MR-G3-type methods defined by Eqs. 5, 6, and 7, 3 4

4 8760 J. Chem. Phys., Vol. 115, No. 19, 15 November 2001 So lling et al. TABLE III. MR-G2 MP2,SVP heat of formation, ionization energies, electron affinities, and proton affinities. Values in parentheses are the differences between experimental and MR-G2 MP2,SVP values. a Heats of LiH PH F formation BeH H 2 S CO CH HCl Na CH 3 2 B Li Si CH 1 2 A LiF P CH C 2 H S CH C 2 H Cl NH CN NaCl NH HCN SiO NH CO CS OH HCO SO H 2 O HCHO ClO HF CH 3 OH ClF SiH 1 2 A N CH 3 Cl SiH 3 2 B N 2 H CH 3 SH SiH NO HOCl SiH O SO PH H 2 O Ionization Li Cl HCl energies Be CH C 2 H B NH C 2 H C OH CO N OH N 2 2 g O HF N 2 2 u F SiH O Na PH P Mg PH S Al PH Cl Si SH ClF P H 2 S 2 B CS S H 2 S 2 A Electron C CH SH affinities O NH O F NH NO Si OH CN P SiH PO S SiH S Cl SiH Cl CH PH CH PH Proton NH SiH HCl affinities H 2 O PH C 2 H H 2 S a Values in kcal mol 1. The heats of formation are 298 K values, whereas the remaining quantities refer to 0 K. E MR-G2 MP2,SVP E MRCI Q/6-31G d E CASPT2/6-311 G 3df,2p E CASPT2/6-31G d ZPVE HLC, 5 E MR-G3 MP2 E MRCI Q/6-31G d E CASPT2/G3MP2large E CASPT2/6-31G d E SO ZPVE HLC. 7 E MR-G2 MP2 The spin orbit correction E(SO) used in our MR- E MRCI Q/6-311G d,p G3 MP2 calculations Eq. 7 is the same as that used in E CASPT2/6-311 G 3df,2p E CASPT2/6-311G d,p ZPVE HLC, 6 G3 theory. 4 In order to investigate the additivity assumptions in Eqs. 5 7, we have also constructed the multireference equiva-

5 J. Chem. Phys., Vol. 115, No. 19, 15 November 2001 Multireference G2 and G TABLE IV. MR-G2 MP2 heats of formation, ionization energies, electron affinities, and proton affinities. Values in parentheses are the differences between experimental and MR-G2 MP2 values. a Heats of LiH PH F formation BeH H 2 S CO CH HCl Na CH 3 2 B Li Si CH 1 2 A LiF P CH C 2 H S CH C 2 H Cl NH CN NaCl NH HCN SiO NH CO CS OH HCO SO H 2 O HCHO ClO HF CH 3 OH ClF SiH 1 2 A N CH 3 Cl SiH 3 2 B N 2 H CH 3 SH SiH NO HOCl SiH O SO PH H 2 O Ionization Li Cl HCl energies Be CH C 2 H B NH C 2 H C OH CO N H 2 O N 2 2 g O HF N 2 2 u F SiH O Na PH P Mg PH S Al PH Cl Si SH ClF P H 2 S 2 B CS S H 2 S 2 A Electron C CH SH affinities O NH O F NH NO Si OH CN P SiH PO S SiH S Cl SiH Cl CH PH CH PH Proton NH SiH HCl affinities H 2 O PH C 2 H H 2 S a Values in kcal mol 1. The heats of formation are 298 K values, whereas the remaining quantities refer to 0 K. lents of the G2/QCI method 3 b,21 and its G3 analog 22 Eqs. 8 and 9, E MR-G2/MRCI Q E MRCI Q/6-311 G 3df,2p ZPVE HLC, E MR-G3/MRCI Q 8 E MRCI Q/G3MP2large E SO ZPVE HLC. MP2, MR-G2/MP2, and MR-G3/MP2. 22 For example, the multireference versions correspond to large-basis-set CASPT2 calculations, and E MR-G2/MP2 E CASPT2/6-311 G 3df,2p ZPVE HLC 10 9 E MR-G3/MP2 E CASPT2/G3MP2large E SO In a third set of calculations, we have investigated the performance of pure MP2 and MR-MP2 analogs of G2- and G3-type procedures, denoting such methods as G2/MP2, G3/ ZPVE HLC. The single-reference analogs are obtained as 11

6 8762 J. Chem. Phys., Vol. 115, No. 19, 15 November 2001 So lling et al. TABLE V. MR-G3 MP2 heats of formation, ionization energies, electron affinities, and proton affinities. Values in parentheses are the differences between experimental and MR-G3 MP2 values. a Heats of LiH PH F formation BeH H 2 S CO CH HCl Na CH 3 2 B Li Si CH 1 2 A LiF P CH C 2 H S CH C 2 H Cl NH CN NaCl NH HCN SiO NH CO CS OH HCO SO H 2 O HCHO ClO HF CH 3 OH ClF SiH 1 2 A N CH 3 Cl SiH 3 2 B N 2 H CH 3 SH SiH NO HOCl SiH O SO PH H 2 O Ionization Li Cl HCl energies Be CH C 2 H B NH C 2 H C OH CO N H 2 O N 2 2 g O HF N 2 2 u F SiH O Na PH P Mg PH S Al PH Cl Si SH ClF P H 2 S 2 B CS S H 2 S 2 A Electron C CH SH affinities O NH O F NH NO Si OH CN P SiH PO S SiH S Cl SiH Cl CH PH CH PH Proton NH SiH HCl affinities H 2 O PH C 2 H H 2 S a Values in kcal mol 1. The heats of formation are 298 K values whereas the remaining quantities refer to 0 K. E G2/MP2 E MP2/6-311 G 3df,2p ZPVE HLC 12 E MR QD -G2 MP2,SVP E MRCI Q/6-31G d and E G3/MP2 E MP2/G3MP2large E SO ZPVE HLC. 13 Finally, we have examined for five of the methods, the effect of using MCQDPT2 in place of CASPT2. For example, MR QD -G2 MP2,SVP is defined by E MCQDPT2/6-311 G 3df,2p E MCQDPT2/6-31G d ZPVE HLC. 5a Similar definitions apply to MCQDPT2 analogs of MR- G2 MP2, MR-G3 MP2, MR-G2/MP2, and MR-G3/MP2. Unless otherwise noted, all energy calculations were carried out using MOLPRO 96 Ref. 23 a and MOLPRO b MOLPRO is currently the most efficient code available for such calculations. The MCQDPT2 calculations were per-

7 J. Chem. Phys., Vol. 115, No. 19, 15 November 2001 Multireference G2 and G TABLE VI. Comparison of the mean absolute deviations kcal mol 1 from experimental data for multi- and single-reference G2 and G3-type methods. a Test set H f IE EA PA Total Number of comparisons G2 MP2,SVP MR-G2 MP2,SVP MR QD -G2 MP2,SVP G2 MP MR-G2 MP MR QD -G2 MP G3 MP MR-G3 MP MR QD -G3 MP G2/QCI b MR-G2/MRCI Q MR-G3/MRCI Q G2/MP MR-G2/MP MR QD -G2/MP G3/MP MR-G3/MP MR QD -G3/MP a Unless otherwise noted, all data refer to the 123 energy test set. In the case of the standard Gn methods, this involved reoptimization of the HLC parameters for the reduced set, leading to results that differ slightly from published values based on the full G2-1 test set Refs. 3 and 4. b Data obtained from Ref. 3 b and refer to the full 125 energy G2-1 test set. TABLE VII. Comparison of mean absolute deviations MAD, kcal mol 1 from experimental data for multi- and single-reference G2 and G3-type methods. a Method MAD Method MAD G2 MP2.SVP 1.60 MR-G2 MP2.SVP 1.41 G2 MP MR-G2 MP G3 MP MR-G3 MP G2/QCI b 1.17 MR-G2/MRCI Q 1.84 G3/QCI MR-G3/MRCI Q 1.58 G2/MP Mr-G2/MP G3/MP MR-G3/MP a All data refer to the 123 molecule test set. In the case of the standard Gn methods, this involved reoptimization of the HLC parameters for the reduced set, leading to results that differ slightly from published values based on the full G2-1 test Refs. 3, 4. b Data obtained from Ref. 3 b and refer to the full 125 energy G2-1 test set. give the smallest mean absolute deviation from experiment for the 123 energy comparisons in our slightly reduced G2-1 test set. We have employed the same minimization procedure as Curtiss, 25 and we are able to reproduce the higher level correction and the mean absolute deviation reported by Curtiss et al. for the G2 MP2 method from the raw electronic energies. 3 c The optimized A parameters for the various G2- type methods are listed in Table I. formed with GAMESS. 18 The total energies for all the systems investigated in the present study, as well as the MCQDPT2 tables of relative energies, are available as an EPAPS document Tables S-I to S-XIX. 24 B. The higher level correction The G2 and G3 methods involve different forms of higher level corrections. The derivation of the parameters involved in the G2 and G3 methods are therefore discussed separately below. The justification and possible problems associated with the use of the higher level correction have previously been discussed by Pople et al The test set A slightly reduced version of the G2-1 test set 3 a was used to obtain the higher-level-correction parameters and to assess the performance of the various methods. The reduced set includes 123 of the 125 energy comparisons of the standard G2-1 set. The heats of formation of ethane and disilane were omitted because the 14, 14 full-valence active space in these two cases makes the MR-CI calculations computationally too demanding. 3. The G3 higher level correction In the G3 method, there are separately optimized higherlevel-correction terms for molecules and atoms. They have the form shown in Eq. 15 molecules and Eq. 16 atoms, HLC An B n n, 15 HLC Cn D n n. 16 We have used the same form of the HLC for all the G3-type methods examined here. The A, B, C, and D parameters are all obtained by minimization of the mean absolute deviation between experiment and theory for the 123 thermochemical quantities in the reduced G2-1 test set. Again, we have employed the same minimization procedure as Curtiss. 25 The optimized parameters for the six G3-type methods are listed in Table II. Starting from the raw electronic energies of the 299 energies in the entire G2/97 test set, our procedure reproduces for both G3 MP2 and G3 the higher-levelcorrection parameters and the mean absolute deviations reported by Curtiss et al. 4 a,4 b III. RESULTS AND DISCUSSION 2. The G2 higher level correction The G2 higher level correction has the form shown in Eq. 14, where n and n are the number of and valence electrons, respectively, Having optimized the higher-level-correction parameters for 12 different MR-Gn procedures as well as two related single-reference Gn procedures, we are now in a position to assess their performance. Thermochemical properties that are examined include heats of formation ( H 0 f ), ionization energies HLC An Bn. 14 IE, electron affinities EA, and proton affinities PA. We have used this form in all the G2-type methods examined in the present study. The B parameter is constrained to be 0.19 mhartrees in all cases so as to give the correct energy for the hydrogen atom, while the A parameter is chosen to A. MR-G2 MP2,SVP, MR-G2 MP2, and MR-G3 MP2 Relative energies calculated at the MR-G2 MP2,SVP, MR-G2 MP2, and MR-G3 MP2 levels are presented in

8 8764 J. Chem. Phys., Vol. 115, No. 19, 15 November 2001 So lling et al. TABLE VIII. MR-G2/MRCI Q a heats of formation, ionization energies, electron affinities, and proton affinities. Values in parentheses are the differences between experimental and MR-G2/MRCI Q values. b Heats of LiH PH F formation BeH H 2 S CO CH HCl Na CH 3 2 B Li Si CH 1 2 A LiF P CH C 2 H S CH C 2 H Cl NH CN NaCl NH HCN SiO NH CO CS OH HCO SO H 2 O HCHO ClO HF CH 3 OH ClF SiH 1 2 A N CH 3 Cl SiH 3 2 B N 2 H CH 3 SH SiH NO HOCl SiH O SO PH H 2 O Ionization Li Cl HCl energies Be CH C 2 H B NH C 2 H C OH CO N H 2 O N 2 2 g O HF N 2 2 u F SiH O Na PH P Mg PH S Al PH Cl Si SH ClF P H 2 S 2 B CS S H 2 S 2 A Electron C CH SH affinities O NH O F NH NO Si OH CN P SiH PO S SiH S Cl SiH Cl CH PH CH PH Proton NH SiH HCl affinities H 2 O PH C 2 H H 2 S a Corresponding to MRCI Q/6-311 G(3df,2p) ZPVE HLC. b Values in kcal mol 1. The heats of formation are 298 K values whereas the remaining quantities refer to 0 K. Tables III V, while a statistical analysis, including a comparison with corresponding single-reference SR methods, is shown in Tables VI and VII. Examination of Table VI shows that, in comparison with the corresponding single-reference methods, results for the three MR procedures are all slightly worse for heats of formation, significantly better for ionization energies and electron affinities, and significantly worse for proton affinities. The overall mean absolute deviations MADs are quite similar for MR-G2 MP2 compared with G2 MP2 and for MR-G3 MP2 compared with G3 MP2. However, MR- G2 MP2,SVP produces better overall results than G2 MP2,SVP. Of the 123 energy comparisons, there are 10 cases of deviations of 3 kcal mol 1 for MR-G2 MP2,SVP, 15 cases for MR-G2 MP2, and 11 cases for MR-G3 MP2. Of these, eight are common to all three methods: the heats of formation of Li 2,Na 2, and SO 2, the ionization energies of Be, Na, and S, the electron affinity of CH 3 and the proton affinity of H 2 O. In comparison, there are 21 cases of deviations of 3 kcal mol 1 for G2 MP2,SVP, 12 cases for G2 MP2, and 10 cases for G3 MP2. Six of the eight deviant cases with the MR-Gn methods are also poor with G3 MP2 : the heats of formation of Li 2,Na 2, and SO 2, and the ionization energies of Be, Na, and S. The poor results obtained for SO 2 in G2-type calculations have been shown previously by Martin 26 to be the result of inadequate basis sets.

9 J. Chem. Phys., Vol. 115, No. 19, 15 November 2001 Multireference G2 and G TABLE IX. MR-G3/MRCI Q a heats of formation, ionization energies, electron affinities, and proton affinities. Values in parentheses are the differences between experimental and MR-G3/MRCI Q values. b Heats of LiH PH F formation BeH H 2 S CO CH HCl Na CH 3 2 B Li Si CH 1 2 A LiF P CH C 2 H S CH C 2 H Cl NH CN NaCl NH HCN SiO NH CO CS OH HCO SO H 2 O HCHO ClO HF CH 3 OH ClF SiH 1 2 A N CH 3 Cl SiH 3 2 B N 2 H CH 3 SH SiH NO HOCl SiH O SO PH H 2 O Ionization Li Cl HCl energies Be CH C 2 H B NH C 2 H C OH CO N H 2 O N 2 2 g O HF N 2 2 u F SiH O Na PH P Mg PH S Al PH Cl Si SH ClF P H 2 S 2 B CS S H 2 S 2 A Electron C CH SH affinities O NH O F NH NO Si OH CN P SiH PO S SiH S Cl SiH Cl CH PH CH PH Proton NH SiH HCl affinities H 2 O PH C 2 H H 2 S a Corresponding to MRCI Q/G3MP2large ZPVE HLC. b Values in kcal mol 1. The heats of formation are 298 K values, whereas the remaining quantities refer to 0 K. Notably poorer performance by the MR procedures compared with SR is observed for the electron affinity of CH 3, and for the proton affinities of NH 3 and H 2 O. The electron affinity of CH 3 is calculated to be negative by all the MR procedures, in contrast to the SR methods that all correctly predict a positive electron affinity. Likewise, the proton affinities of NH 3 and H 2 O are consistently poorly predicted by the MR procedures. In fact, because the errors for NH 3 and H 2 O are of opposite sign, the error in the protontransfer reaction between H 3 O and NH 3 is a substantial kcal mol 1. In contrast, the corresponding SR procedures predict this proton-transfer energy with an accuracy of kcal mol 1. MR-G3 MP2 performs best of the MR methods examined in this study. It gives results comparable to those of G3 MP2 for heats of formation, ionization energies, and electron affinities, but much poorer results for proton affinities. There are significant improvements for a small number of cases for which G3 MP2 gives larger errors: the heat of formation of CS, the ionization energy of O 2, and the electron affinities of C, O, and NH. B. MR-G2ÕMRCI Q and MR-G3ÕMRCI Q The MR-G2 MP2,SVP, MR-G2 MP2, and MR- G3 MP2 procedures aim to approximate the results of MRCI Q/6-311 G(3df,2p) or MRCI Q/G3MP2large calculations together with HLC and ZPVE corrections by

10 8766 J. Chem. Phys., Vol. 115, No. 19, 15 November 2001 So lling et al. TABLE X. G2/MP2 a heats of formation, ionization energies, electron affinities, and proton affinities. Values in parentheses are the differences between experimental and G2/MP2 values. b Heats of LiH PH F formation BeH H 2 S CO CH HCl Na CH 3 2 B Li Si CH 1 2 A LiF P CH C 2 H S CH C 2 H Cl NH CN NaCl NH HCN SiO NH CO CS OH HCO SO H 2 O HCHO ClO HF CH 3 OH ClF SiH 1 2 A N CH 2 Cl SiH 3 2 B N 2 H CH 3 SH SiH NO HOCl SiH O SO PH H 2 O Ionization Li Cl HCl energies Be CH C 2 H B NH C 2 H C OH CO N H 2 O N 2 2 g O HF N 2 2 u F SiH O Na PH P Mg PH S Al PH Cl Si SH ClF P H 2 S 2 B CS S H 2 S 2 A Electron C CH SH affinities O NH O F NH NO Si OH CN P SiH PO S SiH S Cl SiH Cl CH PH CH PH Proton NH SiH HCl affinities H 2 O PH C 2 H H 2 S a Corresponding to MP2/6-311 G(3df,2p) ZPVE HLC. b Values in kcal mol 1. The heats of formation are 298 K values, whereas the remaining quantities refer to 0 K. assuming the additivity of correlation and basis set effects. It is of interest to examine the reliability of such additivity approximations by carrying out the large-basis-set MRCI Q calculations explicitly. This gives rise to the MR-G2/MRCI Q and MR-G3/MRCI Q procedures, defined by Eqs. 8 and 9, which are analogous to the G2/QCI procedure examined previously. 3 b Relative energies at the MR-G2/MRCI Q and MR-G3/MRCI Q levels are presented in Tables VIII and IX, with statistical summaries included in Tables VI and VII. Quite unexpectedly, MR-G2/MRCI Q and MR-G3/MRCI Q show larger overall deviations from experiment than MR-G2 MP2 and MR-G3 MP2, respectively. This means that the additivity approximations in Eqs. 6 and 7 are actually helpful in improving the results which become worse if the additivity is removed, which must surely be a fortuitous situation. There are now 23 (MR-G2/MRCI Q) and 15 (MR-G3/MRCI Q) cases for which the deviations from experiment exceed 3 kcal mol 1. The MR-Gn/MRCI Q procedures perform significantly worse than the corresponding standard MR-Gn methods for heats of formation and electron affinities, slightly worse for proton affinities and comparably for ionization energies. Significantly larger errors compared with standard MR-Gn are observed for both MR-G2/MRCI Q and MR-G3/MRCI Q for the heats of formation of SiH 2 ( 1 A 1 ), SiH 4, and SO 2. In addition, MR-G2/MRCI Q shows large errors for the heats of formation of S 2,Cl 2, SO, and ClO,

11 J. Chem. Phys., Vol. 115, No. 19, 15 November 2001 Multireference G2 and G TABLE XI. G3/MP2 a heats of formation, ionization energies, electron affinities, and proton affinities. Values in parentheses are the differences between experimental and G3/MP2 values. b Heats of LiH PH F Formation BeH H 2 S CO CH HCl Na CH 3 2 B Li Si CH 1 2 A LiF P CH C 2 H S CH C 2 H Cl NH CN NaCl NH HCN SiO NH CO CS OH HCO SO H 2 O HCHO ClO HF CH 3 OH ClF SiH 1 2 A N CH 3 Cl SiH 3 2 B N 2 H CH 3 SH SiH NO HOCl SiH O SO PH H 2 O Ionization Li Cl HCl energies Be CH C 2 H B NH C 2 H C OH CO N H 2 O N 2 2 g O HF N 2 2 u F SiH O Na PH P Mg PH S Al PH Cl Si SH ClF P H 2 S 2 B CS S H 2 S 2 A Electron C CH SH affinities O NH O F NH NO Si OH CN P SiH PO S SiH S Cl SiH Cl CH PH CH PH Proton NH SiH HCl affinities H 2 O PH C 2 H H 2 S a Corresponding to MP2/G3MP2large E(SO) ZPVE HLC. b Values in kcal mol 1. The heats of formation are 298 K values, whereas the remaining quantities refer to 0 K. while MR-G3/MRCI Q performs poorly for NaCl. Both MR-G2/MRCI Q and MR-G3/MRCI Q significantly underestimate electron affinities, with a noticeable deterioration in the predictions for O, F, OH, O 2, NO, and PO. The proton affinities of NH 3 and H 2 O continue to be poorly predicted. C. G2ÕMP2 and G3ÕMP2 The G2/QCI procedure 3 b obtains relative energies on the basis of QCISD T /6-311 G(3df,2p) calculations together with ZPVE and HLC corrections. It is of interest to see how the corresponding MP2 calculations fare. With this in mind, we have analyzed results corresponding to MP2/6-311 G(3df,2p) ZPVE HLC and MP2/G3MP2 large ZPVE HLC. These procedures are designated G2/ MP2 and G3/MP2, respectively, and are defined by Eqs. 12 and 13. Calculated relative energies are presented in Tables X and XI, with statistical summaries again included in Tables VI and VII. It is immediately clear from Tables VI and VII that G2/ MP2 and G3/MP2 are not particularly useful levels of theory from the viewpoint of thermochemical reliability. The mean absolute deviations are 4.19 and 3.36 kcal mol 1, with 69 and 53, respectively, out of the 123 energy comparisons showing deviations exceeding 3 kcal mol 1. The only area where the errors are modest is for proton affinities. The G2/

12 8768 J. Chem. Phys., Vol. 115, No. 19, 15 November 2001 So lling et al. TABLE XII. MR-G2/MP2 a heats of formation, ionization energies, electron affinities, and proton affinities. Values in parentheses are the differences between experimental and MR-G2/MP2 values. b Heats of LiH PH F formation BeH H 2 S CO CH HCl Na CH 3 2 B Li Si CH 1 2 A LiF P CH C 2 H S CH C 2 H Cl NH CN NaCl NH HCN SiO NH CO CS OH HCO SO H 2 O HCHO ClO HF CH 3 OH ClF SiH 1 2 A N CH 3 Cl SiH 3 2 B N 2 H CH 3 SH SiH NO HOCl SiH O SO PH H 2 O Ionization Li Cl HCl energies Be CH C 2 H B NH C 2 H C OH CO N H 2 O N 2 2 g O HF N 2 2 u F SiH O Na PH P Mg PH S Al PH Cl Si SH ClF P H 2 S 2 B CS S H 2 S 2 A Electron C CH SH affinities O NH O F NH NO Si OH CN P SiH PO S SiH S Cl SiH Cl CH PH CH PH Proton NH SiH HCl affinities H 2 O PH C 2 H H 2 S a Corresponding to CASPT2/6-311 G(3df,2p) ZPVE HLC. b Values in kcal mol 1. The heats of formation are 298 K values whereas the remaining quantities refer to 0 K. MP2 and G3/MP2 methods are not recommended for general use. D. MR-G2ÕMP2 and MR-G3ÕMP2 The multireference analogs of G2/MP2 and G3/MP2 use large-basis-set MR-MP2 specifically CASPT2 calculations together with ZPVE and HLC corrections. They are designated MR-G2/MP2 and MR-G3/MP2 and are defined by Eqs. 10 and 11, respectively. Results are presented in Tables XII and XIII. Examination of the statistical summaries in Tables VI and VII shows a number of interesting points. In the first place, MR-G2/MP2 and MR-G3/MP2 perform significantly better than G2/MP2 and G3/MP2, with MADs of 2.01 and 1.66 kcal mol 1 compared with 4.19 and 3.36 kcal mol 1. They are only slightly worse than MR-G2/MRCI Q and MR-G3/MRCI Q MADs of 1.84 and 1.58 kcal mol 1, respectively. However, they do not perform as well as the standard MR-Gn procedures. For example, the MADs are larger than with MR-Gn(MP2) for virtually all the thermochemical properties in Table VI. There are 31 MR-G2/MP2 and 23 MR-G3/MP2 out of 123 energy comparisons for which the error exceeds 3 kcal mol 1. Large errors occur for most of the systems that were noted in connection with the standard MR-Gn procedures. However, there are additional cases for which there

13 J. Chem. Phys., Vol. 115, No. 19, 15 November 2001 Multireference G2 and G TABLE XIII. MR-G3/MP2 a heats of formation, ionization energies, electron affinities, and proton affinities. Values in parentheses are the differences between experimental and MR-G3 MP2 /MP2 values. b Heats of LiH PH F formation BeH H 2 S CO CH HCl Na CH 3 2 B Li Si CH 1 2 A LiF P CH C 2 H S CH C 2 H Cl NH CN NaCl NH HCN SiO NH CO CS OH HCO SO H 2 O HCHO ClO HF CH 3 OH ClF SiH 1 2 A N CH 3 Cl SiH 3 2 B N 2 H CH 3 SH SiH NO HOCl SiH O SO PH H 2 O Ionization Li Cl HCl energies Be CH C 2 H B NH C 2 H C OH CO N H 2 O N 2 2 g O HF N 2 2 u F SiH O Na PH P Mg PH S Al PH Cl Si SH ClF P H 2 S 2 B CS S H 2 S 2 A Electron C CH SH affinities O NH O F NH NO Si OH CN P SiH PO S SiH S Cl SiH Cl CH PH CH PH Proton NH SiH HCl affinities H 2 O PH C 2 H H 2 S a Corresponding to CASPT2/G3MP2large E(SO) ZPVE HLC. b Values in kcal mol 1. The heats of formation are 298 K values, whereas the remaining quantities refer to 0 K. are noticeable errors. In the case of heats of formation, CO 2, SiO, NH 3, and O 2 now show significant errors, and the error for SO 2 has moved from large negative to small positive. The MR-Gn/MP2 ionization energies are significantly worse than corresponding MR-Gn(MP2) values, and significant errors now occur at both MR-Gn/MP2 levels for the additional systems B, NH 3, and O 2. In the case of electron affinities, there are very large deviations for CH and 5.9 kcal mol 1, respectively, and F, PO, and Cl 2 also have large errors. The errors in the H 3 O /NH 3 proton-transfer energy are now kcal mol 1. Despite these shortcomings, the MR-G3/MP2 procedure may prove useful in situations for which single-reference methods are inadequate, especially since the use of large active spaces is more limiting for MR-CI than for MR-MP2 methods. E. MCQDPT2 vs CASPT2 Our default MR-MP2 method is the CASPT2 procedure of the MOLPRO suite of programs. 15,23 However, it is of interest to see how the alternative MCQDPT2 procedure that is available in the GAMESS program 18 compares. Results analogous to those of Tables III, IV, V, XII, and XIII are available as an EPAPS document. 24 Statistical summaries are included in Table VI. The general observation is that the CASPT2-based re-

14 8770 J. Chem. Phys., Vol. 115, No. 19, 15 November 2001 So lling et al. TABLE XIV. MR-Gn and Gn timings. a Method NH 3 C 2 H 2 HCHO MR-G3/MP MR-G2/MP MR-G3 MP MR-G2 MP2,SVP MR-G2 MP MR-G3/MRCI Q MR-G2/MRCI Q G3/MP G2/MP G3 MP G2 MP2,SVP G2 MP G3/QCI G2/QCI a In seconds using MOLPRO 98 on a single processor of a VPP300 with 1700 Mb memory. The active-space sizes for NH 3,C 2 H 2, and HCHO are 8,7, 10,10, and 12,10 for NH 3, C 2 H 2, and HCHO, respectively. The Gn timings refer to Gn(CCSD) calculations, i.e., in which CCSD T is used in place of QCISD T. sults and MCQDPT2-based results are normally very similar. With MR-G2 MP2,SVP, there are nine cases where the difference lies between 1 and 3 kcal mol 1 and just one case NaCl where the difference is greater than 3 kcal mol 1. With MR-G2 MP2, there are eight cases where the difference lies between 1 and 3 kcal mol 1, two cases NaCl and CH 3 SH where the difference lies between 3 and 5 kcal mol 1, and one case (Cl 2 ) where the difference exceeds 5 kcal mol 1. For MR-G3 MP2, there are five cases where the difference lies between 1 and 3 kcal mol 1 and one case NaCl where the difference exceeds 3 kcal mol 1. The differences are larger with MR-G2/MP2 and MR-G3/MP2, with 33 and 30 cases, respectively, of differences lying between 1 and 3 kcal mol 1, and two and three cases, respectively, of differences exceeding 3 kcal mol 1. Although the differences between the results of the CASPT2-based methods and MCQDPT2-based methods are relatively small, it may be seen from Table VI that the CASPT2-based methods virtually always perform slightly better statistically. TABLE XV. Comparison of the ten largest deviations between experiment and the values calculated by MR-G3 MP2 and G3 MP2. a Quantity MR-G3 MP2 b Quantity G3 MP2 c H 0 f (SO 2 ) IE Be IE Be EA NH H 0 f (Na 2 ) IE O EA CH H 0 f (SO 2 ) PA H 2 O IE S PA NH 3 ) EA C IE Na EA O H 0 f (C 2 H 4 ) H 0 f (Na 2 ) H 0 f (NaCl) H 0 f (CS) IE S IE Na a In kcal mol 1. b Values in parentheses are the corresponding G3 MP2 deviations from Ref. 4 b. c Taken from Ref. 4 b ; values in parentheses are the corresponding MR- G3 MP2 deviations from the present work. in the MR-G2/MP2 and MR-G3/MP2 methods, leads to a substantial reduction in CPU time. The cost of the standard SR Gn methods goes up much more slowly than the MR methods. We should emphasize that in this initial implementation of MR-Gn procedures, we uniformly use a fullvalence active space and this leads to the very rapid increase in computational cost with size of molecule. Clearly this will be modified in implementations that use smaller active spaces. The relative costs of the MR methods for the larger active spaces follow the pattern, 27 MR-G2/MP2 MR-G3/MP2 MR-G2 MP2,SVP MR-G3 MP2 MR-G2/MRCI Q MR-G3/MCRCI Q. 17 There is a large increase in CPU time in going from MR-G3/ MP2 to MR-G3 MP2 but a much smaller further increase in going to MR-G3/MRCI Q. The single-reference methods show the same pattern, G2/MP2 G3/MP2 G2 MP2,SVP G3 MP2 G2/QCI G3/QCI. 18 F. Timing comparisons and additional comments Because the choice of method in quantum chemistry studies often involves a compromise between accuracy and computational expense, it is important to examine the relative timings of the various MR-Gn procedures introduced in the present article and to make comparisons with corresponding standard single-reference Gn methods. It should be emphasized that the timings depend on many factors and so the present data are intended largely to enable qualitative conclusions to be drawn. We can see from Table XIV that for active spaces of up The MR-G2/MRCI Q and MR-G3/MRCI Q procedures are the most demanding of the methods investigated in the present work in terms of both memory and CPU usage. However, our results show that these two methods are by no means the most accurate. This is not an overly comforting situation, since the aim of the other methods that we have examined is to approximate their large-basis-set MRCI Q counterparts by means of additivity. It turns out that the MR-G2 and MR-G3 schemes that we have devised do not succeed very well in mimicking the MR-Gn/MRCI Q results. This state of affairs fortuitously results in cheaper methods MR-G2 MP2,SVP and MR-G3 MP2 that are to about 8 orbitals, all methods are very cheap. For 10 and 12 more accurate than their more expensive counterparts active orbitals, the times increase rapidly for the MR-Gn procedures that involve MR-CI calculations, while for 14 and 16 active orbitals, such procedures are starting to become intractable. Elimination of the MR-CI component, as (MR-G2/MRCI Q and MR-G3/MRCI Q. The two MR methods for which the correlation correction is based on the 6-31G(d) split-valence basis set MR- G2 MP2,SVP and MR-G3 MP2 give the most accurate